GOAL: Solve two new types of problems: Proportion problems and Variation problems.
A proportion is when two ratios are equal and can be solved with cross-multiplication. Note: Students
use the term cross-multiply all the time in the wrong instance. This is the only correct use of the term.
EX: Solve the following proportion
Cross multiply
(
)
( )
Check the original equation….Does
cause a “divide by zero problem?”
EX: 80 people in a particular neighborhood are surveyed, and 35 are found to be home owners. If the
neighborhood has 3100 people in it, use a proportion to determine the number of home owners in the
neighborhood.
(
)
(
)
EX: A contractor is building several kitchens for a large building. $500 worth of tile is sufficient to tile 3
kitchens. How much more will it cost to build a total of 5 kitchens?
( )
(
)
They will need an additional $333.33 dollars to complete 5 kitchens.
EX: A map has a legend that says
. If the distance between two points on a map
measures 5.6 inches, how many miles are two locations apart in reality?
(
)
(
)
The two locations are about 7.4667 miles apart.
Direct and Inverse Variation
Variation describes a relationship between two quantities. The relationship involves a number called
the constant of variation (we often use the letter k).


Direct Variation
o If varies directly with :
Then
o Usually means when one value gets bigger, then so does the other value.
Inverse Variation
o If varies inversely with :
Then
⁄
o Usually means when one value gets bigger, then the other gets smaller.
What are we to do?
 We are usually given two quantities that go together (
)
 It is our job to set up an equation to find
.
 We then often have to use this information to answer a question.
EX: Answer the following questions about variation.
1) The amount of time a faucet is dripping varies directly with how much water has leaked out. A
faucet has been leaking for 9 hours and has leaked 2 gallons of water.
a. Find the constant of variation (use )
b. Write a model for the situation
c. Use the model to find out how much water will be leaked in 24 hours.
a. Let’s call time and the amount of water leaked
water…so….
They give me some information
( ) So
. It says time varies directly with
b. The model is now
c. Use the new time given to find the new amount of water leaked.
( )
Multiply by 2
About 5.333 gallons will have leaked in 24 hours.
2) The speed a jet ski can travel varies inversely with its width. A jet ski with a width of 40 inches
can go 30mph.
a. Find the constant of variation (use )
b. Write a model for the situation
c. Use the model to find out how fast a jet ski can go if its width is 36 inches.
a. Let’s use for speed, and for width. It says speed varies inversely with width…so
⁄
Use the information given to find
so multiplying by 40 gives us
b. The model is now
⁄
c. Use the new information given.
So the slightly slimmer jet ski can go 33.333 miles per hour.
3) The force of gravity between two objects varies inversely with the distance between them (not
exactly). Two objects have 20 pounds of force between them when they are separated by 5
kilometers.
a. Find the constant of variation (use )
b. Write a model for the situation
c. Use the model to find out how far apart they must be for the force to be 25 pounds.
a. Use to mean force, and to mean distance. It says force varies inversely with
distance.
⁄
so using the initial information…
b.
c.
Multiply both sides by
to get it out of the denominator.
So the objects must be 4 kilometers apart to experience 25 pounds of force.
4) The GPA of a student at college varies directly with the number of hours spent studying during a
semester. A student received a 2.8 when they studying for a total of 80 hours over the
semester.
a. Find the constant of variation (use )
b. Write a model for the situation
c. Use the model to determine how much they must study to raise their GPA to a 3.1
a. Let
represent GPA, and let
represent time spent studying.
so
b.
c.
So the student will have to study for 88.57 hours to raise their GPA to a 3.1.
WARNING: Proportions (setting two ratios equal) can be used to solve problems
involving direct variation, but inverse variation problems cannot!
5) The amount of medication “M” in the blood stream varies inversely with the time since it has
been administered. There is 20 ppm (parts per million) of medication M in a patient’s blood
stream 48 hours after it has been administered.
a. Find the constant of variation (use )
b. Write a model for the situation
c. Use the model to determine when there will only be 10 ppm of medication M in the
patient’s blood stream.
a. Let represent the amount of medication and represent the amount of time which
has passed. The problem says varies inversely with so..
⁄
Multiplying by 48 on both sides gives us…
b.
c.
multiply by
on both sides to get it out of the denominator…
It will take 96 hours for the medication be at a level of 10 ppm.